Tu ning wa es and b eakdown o
incomp essible lows
Angel Cas oa, Diego Có dobaa, Cha les L. Fe e manb,1, F ancisco Gancedoc, and Ma ía López-Fe nándezd
aIns i u o de Ciencias Ma emá icas, Consejo Supe io de In es igaciones Cien í icas, Se ano 123, 28006 Mad id, Spain; bDepa men o Ma hema ics,
P ince on Uni e si y, 1102 Fine Hall, Washing on Road, P ince on, NJ 08544; cDepa men o Ma hema ics, Uni e si y o Chicago, 5734 Uni e si y A enue,
Chicago, IL 60637; and dIns i u ü Ma hema ik, Uni e si ä Zü ich Win e hu e s asse 190 CH-8057 Zu ich, Swi ze land
Con ibu ed by Cha les L. Fe e man, Janua y 29, 2011 (sen o e iew No embe 30, 2010)
We conside he e olu ion o an in e ace gene a ed be ween wo
immiscible, incomp essible, and i o a ional luids. Speci ically we
s udy he Muska and wa e wa e p oblems. We show ha s a -
ing wi h a amily o ini ial da a gi en by ðα, 0ðαÞÞ, he in e ace
eaches a egime in ini e ime in which is no longe a g aph. The e-
o e he e exis s a ime whe e he solu ion o he ee bounda y
p oblem pa ame e ized as ðα, ðα, ÞÞ blows up: ‖∂α ‖L∞ð Þ¼∞.In
pa icula , o he Muska p oblem, his esul allows us o each an
uns able egime, o which he Rayleigh–Taylo condi ion changes
sign and he solu ion b eaks down.
1. In oduc ion
He e we s udy wo p oblems o luids mechanics conce ning he
e olu ion o wo incomp essible luids o di e en cha ac e is ics
in 2D. We conside ha bo h luids a e immiscible and o di e -
en cons an densi ies ρ1and ρ2, modeling he dynamics o an
in e ace ha sepa a es he domains Ω1ð Þand Ω2ð Þ. Tha is,
he liquid densi y ρ¼ρðx; Þ;ðx; Þ∈R2×Rþ, is de ined by
ρðx; Þ¼ρ1;x∈Ω1ð Þ
ρ2;x∈Ω2ð Þ¼R2−Ω1ð Þ;[1]
and sa is ies he conse a ion o mass equa ion
ρ þ ·∇ρ¼0;∇· ¼0;[2]
whe e ¼ð 1ðx; Þ; 2ðx; ÞÞ is he eloci y ield. Wi h a ee bound-
a y pa ame e ized by
∂Ωjð Þ¼ zðα; Þ¼ðz1ðα; Þ;z2ðα; ÞÞ:α∈Pg;
we conside open cu es anishing a in ini y
lim
α→∞ðzðα; Þ−ðα;0ÞÞ¼0;
o pe iodic in he space a iable
zðαþ2kπ; Þ¼zðα; Þþ2kπð1;0Þ:
The scala o ici y, ∇⊥· , has he o m
∇⊥· ðx; Þ¼ωðα; Þδðx−zðα; ÞÞ;[3]
i.e., he o ici y is a Di ac measu e on zde ined by
<∇⊥· ;η>¼ZR
ωðα; Þηðzðα; ÞÞdα;
wi h ηðxÞa es unc ion. The sys em is closed by using one o he
ollowing undamen al luid mo ion equa ions:
Da cy’s law
μ
κ ¼−∇p−gρð0;1Þ;[4]
o
Eule equa ions
ρð þ ·∇ Þ¼−∇p−gρð0;1Þ:[5]
He e p¼pðx; Þis p essu e, gg a i y, μ iscosi y, and κpe meabil-
i y o he iso opic medium.
The Muska p oblem (1) is gi en by Eqs. 1,2, and 4, which
conside s he dynamics o wo incomp essible luids o di e en
densi ies h oughou po ous media and Hele–Shaw cells (2, 3). In
his las se ing, he luid is apped be ween wo ixed pa allel
pla es ha a e close enough oge he so ha he luid essen ially
only mo es in wo di ec ions (4).
Taking ρ1¼0, Eqs. 1–3and 5a e known as he wa e wa es
p oblem (see e . 5 and e e ences he ein), modeling he
dynamics o he con ou be ween an in iscid luid wi h densi y
ρ2and acuum (o ai ) unde he in luence o g a i y.
Condi ion 3(deduced by [4], assumed o [5]) allows us o
w i e he e olu ion equa ion in e ms o he ee bounda y as ol-
lows. One could eco e he eloci y ield om [3] by means o
Bio –Sa a law
ðx; Þ¼∇⊥Δ−1ð∇⊥· Þðx; Þ¼ 1
2πZR
ðx−zðα; ÞÞ⊥
jx−zðα; Þj2ωðα; Þdα;
applying he Di ac measu e wi h ampli ude ω. Taking limi s on
he abo e equa ion app oaching he bounda y in he no mal
di ec ion inside Ωj, he eloci y is shown o be discon inuous
in he angen ial di ec ion, bu con inuous in he no mal, and
gi en by he Bi kho –Ro in eg al o he ampli ude ωalong he
in e ace cu e:
BRðz;ωÞðα; Þ¼ 1
2πPV ZR
ðzðα; Þ−zðβ; ÞÞ⊥
jzðα; Þ−zðβ; Þj2ωðβ; Þdβ;
whe e PV deno es p incipal alue. This ac yields he cu e
eloci y om which one can sub ac any e m cin he angen ial
wi hou modi ying he geome y o he in e ace
z ðα; Þ¼BRðz;ωÞðα; Þþcðα; Þ∂αzðα; Þ:[6]
Unde s anding he p oblem as weak solu ions o [1,2, and 4]o
[1–3and 5], he con inui y o he p essu e on he ee bounda y
ollows. The e o e, aking limi s in Da cy’s law om bo h sides
and sub ac ing he esul s in he angen ial di ec ion, i is easy
o close he sys em o Muska (in his pape we conside wo
luids wi h he same iscosi y):
ωðα; Þ¼−ðρ2−ρ1Þκg
μ∂αz2ðα; Þ:[7]
Au ho con ibu ions: A.C., D.C., C.L.F., F.G., and M.L.-F. designed esea ch, pe o med
esea ch, and w o e he pape .
The au ho s decla e no con lic o in e es .
1To whom co espondence should be add essed. E-mail: [email p o ec ed].
4754–4759 ∣PNAS ∣Ma ch 22, 2011 ∣ ol. 108 ∣no. 12 www.pnas.o g/cgi/doi/10.1073/pnas.1101518108
In a simila way o wa e wa es, Eule equa ions yield
ω ðα; Þ¼−2∂ BRðz;ωÞðα; Þ·∂αzðα; Þ−∂
αjωj2
4j∂αzj2ðα; Þ
þ∂αðcωÞðα; Þþ2cðα; Þ∂αBRðz;ωÞðα; Þ·∂αzðα; Þ
−2g∂αz2ðα; Þ:[8]
Then, he wo con ou equa ions a e se by [6and 7] and [6and 8].
Fo hese models, he well-posedness u ns ou o be alse o
some se ings. Rayleigh (6) and Sa man and Taylo (2) ga e a
condi ion ha mus be sa is ied o he linea ized model in o de
o exis a solu ion locally in ime: The no mal componen o he
p essu e g adien jump a he in e ace has o ha e a dis inguished
sign. This quan i y is known as he Rayleigh–Taylo condi ion. I
eads as
σðα; Þ¼−ð∇p2ðzðα; Þ; Þ−∇p1ðzðα; Þ; ÞÞ ·∂⊥
αzðα; Þ>0;
whe e ∇pjðzðα; Þ; Þdeno es he limi g adien o he p essu e
ob ained app oaching he bounda y in he no mal di ec ion inside
Ωjð Þ.
An easy linea iza ion a ound a la con ou ðα; ðα; ÞÞ, allows us
o ind
¼1
2HðωÞ
whe e His he Hilbe ans o m which symbol on he Fou ie
side is gi en by b
H¼−isign ðξÞ. The equa ions
ω¼−ðρ2−ρ1Þκg
μ∂α ; ðlinea Muska Þ
ω ¼2g∂α ; ðlinea wa e wa esÞ
show he pa abolici y o he Muska p oblem when he dense
luid is below (ρ2>ρ1) and he dispe si e cha ac e o wa e
wa es.
1. The e is a wide li e a u e on he Muska p oblem and he
dynamics o wo luids in a Hele–Shaw cell. The e a e wo ks
conside ing he case o a iscosi y jump neglec ing he e ec
o g a i y (7, 8). Local exis ence in a mo e gene al si ua ion
(wi h discon inuous iscosi y and densi y) is shown in e . 9
and also ea ed in e . 10. A di e en app oach o p o e local
exis ence can be ound in e . 11 o he se ing we a e con-
side ing in his pape . The Rayleigh–Taylo s abili y depends
upon he sign o ðρ2−ρ1Þ∂αz1ðα; Þ(11) indica ing ha he
hea ie luid has o be below in he s able case. I he ligh e
luid is below, he p oblem has been shown o be ill-posed
(11). Global-exis ence esul s o small ini ial da a can be
ound in e s. 7 and 11–14. Fo la ge ini ial cu es and pa a-
me e ized by ðα; ðα; ÞÞ, he e a e maximum p inciples o he
L∞and L2no ms o , and decay a es, oge he wi h global
exis ence o Lipschi z cu es i ‖∂α ‖L∞ð0Þ<1(15, 16, 17).
2. The wa e wa es p oblem has been ex ensi ely conside ed (see
e s. 5 and 18 and e e ences he ein). Fo su icien ly smoo h
ee bounda y, he Rayleigh–Taylo condi ion emains posi i e
wi h no bo om conside a ions (19), a ac ha was used o
p o e local exis ence (19). The Rayleigh–Taylo s abili y can
play a di e en ole o he case o non-“almos ”- la bo om
(20). Recen ly, o small ini ial da a, exponen ial ime o
exis ence has been p o en in wo dimensions (21) and global
exis ence in he h ee-dimensional case ( wo-dimensional in-
e ace) (22, 23).
2. Rayleigh–Taylo B eakdown o Muska
This sec ion is de o ed o show he main ing edien s o p o e he
Theo em 2.1. We conside he unc ion
FðzÞðα;βÞ¼ jβj2
jzðαÞ−zðα−βÞj2;α;β∈R;
and in he pe iodic se ing
FðzÞðα;βÞ¼ ‖β‖2
2ðcoshðz2ðαÞ−z2ðα−βÞÞ −cosðz1ðαÞ−z1ðα−βÞÞÞ ;
α;β∈T;[9]
whe e ‖x‖¼dis ðx;2πZÞ.I FðzÞ∈L∞ðR2Þ, hen he cu e zsa-
is ies he a c-cho d condi ion. We say ha he Rayleigh–Taylo
(R-T) o he solu ion o he Muska p oblem b eaks down in ini e
ime i o ini ial da a z0sa is ying σðα;0Þ¼ðρ2−ρ1Þ∂αz1ðα;0Þ>0
he e exis s a ime >0 o which σðα; Þis s ic ly nega i e in a
nonemp y open in e al.
Theo em 2.1. The e exis s a nonemp y open se o ini ial da a in H4,
sa is ying Rayleigh–Taylo and a c-cho d condi ions, o which he
Rayleigh–Taylo condi ion o he solu ion o he Muska p oblem
[1,2, and 4] b eaks down in ini e ime.
A e choosing he app op ia e angen ial e m and a in eg a-
ion by pa s, he con ou equa ion eads
z ðα; Þ¼ρ2−ρ1
2πPV ZR
ðz1ðα; Þ−z1ðβ; ÞÞ
jzðα; Þ−zðβ; Þj2ð∂αzðα; Þ−∂
αzðβ; ÞÞdβ:
Fo a 2πpe iodic in e ace, emo ing he p incipal alue a
in ini y, he equa ion becomes
z ðαÞ¼ðρ2−ρ1Þ
4π
×ZT
sinðz1ðαÞ−z1ðα−βÞÞð∂αzðαÞ−∂
αzðα−βÞÞ
coshðz2ðαÞ−z2ðα−βÞÞ −cosðz1ðαÞ−z1ðα−βÞÞ dβ:
[10]
F om now on, we shall use he pe iodic con igu a ion.
The s eps o he p oo a e as ollows:
1. Fi s , o any ini ial cu e z0ðαÞ¼zðα;0Þin H4 ha sa is y R-T
ðρ2−ρ1Þ∂αz1ðα;0Þ>0
and he a c-cho d condi ion hen he solu ion o he Muska
p oblem zðα; Þbecomes analy ic o 0< <T. Mo eo e ,
zðα; Þis eal analy ic in a s ip
Sð Þ¼ αþiζ:jζj<c g
o ∈ð0;TÞwhe e cdepends only on
in ð0Þ¼in
α
∂αz1ðα;0Þ
j∂αzðα;0Þj2:
The p oo ollows by con olling he quan i ies ex ended on
Sð Þ:
FðzÞðαþiζ;β; Þ
and gðαþiζ; Þby using [9] and o mula
gðα; Þ¼ZT
½sinðz1ðα; Þ−z1ðα−β; ÞÞ∕½coshðz2ðα; Þ
−z2ðα−β; ÞÞ −cosðz1ðα; Þ−z1ðα−β; ÞÞdβ;
espec i ely. The no ms
Cas o e al. PNAS ∣Ma ch 22, 2011 ∣ ol. 108 ∣no. 12 ∣4755
MATHEMATICS
‖FðzÞ‖L∞ðSÞð Þ¼ sup
αþiζ∈Sð Þ;β∈T
jFðzÞðαþiζ;βÞj;
‖z‖2
L2ðSÞð Þ¼∑
ZT
jzðαic ; Þj2dα;
‖z‖2
HjðSÞð Þ¼‖z‖2
L2ðSÞð Þþ∑
ZT
j∂j
αzðαic ; Þj2dα;
o j∈N;
in ð Þ¼ in
αþiζ∈Sð Þ
ℜ∂αz1ðαþiζ; Þ
j∂αzðαþiζ; Þj2:
Then he quan i y
‖z‖2
RTð Þ¼‖z‖2
H4ðSÞð Þþ‖FðzÞ‖L∞ðSÞð Þ
þ1∕ðin ð Þ−c−K∥ℑðgÞ‖H2ðSÞð ÞÞ
sa is ies
d
d ‖z‖RTð Þ≤C‖z‖k
RTð Þ;
o C,K, and kuni e sal cons an s. I yields
‖z‖RTð Þ≤‖z‖RTð0Þ
ð1−C‖z‖k
RTð0Þ Þ1∕k;
p o iding con ol o he analy ici y and T¼1∕ðC‖z‖k
RTð0ÞÞ.
2. Second, he e is a lowe bound on he s ip o analy ici y,
which does no collapse o he eal axis as long as he
Rayleigh–Taylo is g ea e han o equal o 0. Then he e is
a ime Tand a solu ion o he Muska p oblem zðα; Þde ined
o 0< ≤T ha con inues analy ically in o a complex s ip
i ðρ2−ρ1Þ∂αz1≥0, whe e Tis ei he a small cons an o i is
he i s ime a e ical angen appea s, whiche e occu s i s .
We ede ine he s ip
Sð Þ¼ αþiζ:jζj<hð Þ;0<hð0Þg;
and he quan i y ‖z‖2
S¼‖z‖2
H4ðSÞþ‖FðzÞ‖L∞ðSÞwi h his new
Sð Þ. Fo an hð Þdec easing [ he exp ession o hð Þis chosen
la e ], we conside he e olu ion o he mos singula quan i y
∑
Zj∂4
αzðαihð Þ; Þj2dα:
Taking a de i a i e in , one inds
d
d ∑
Zj∂4
αzðαihð ÞÞj2dα≤h0ð Þ
10
∑
ZΛð∂4
αzÞðαihð ÞÞ ·∂4
αzðαihð ÞÞdα
−10h0ð ÞZΛð∂4
αzÞðαÞ·∂4
αzðαÞdα
þ2∑

ℜZ∂4
αz ðαihð ÞÞ ·∂4
αzðαihð ÞÞdα:
Es ima ing in a wise way, one ob ains
d
d ∑
Zj∂4
αzðαihð ÞÞj2dα≤C‖z‖k
Sð Þ
−10h0ð ÞZΛð∂4
αzÞðαÞ·∂4
αzðαÞdαþðC‖z‖k
Sð Þhð Þ
þ1
10 h0ð ÞÞ ZΛð∂4
αzÞðαihð ÞÞ ·∂4
αzðαihð ÞÞdα:
The e o e, choosing
hð Þ¼hð0Þexpð−10CZ
0
‖z‖k
Sð Þd Þ
elimina es he mos dange ous e m. The o he e ms a e ea-
sily con olled, gi ing inally
d
d ∑
Zj∂4
αzðαihð ÞÞj2dα≤C‖z‖kþ2
Sð Þ;
which allows us o each a egime o which he bounda y z
de elops a e ical angen a ime T.
3. Thi d, i is shown he exis ence o a la ge class o analy ic
cu es o which he e exis a poin whe e he angen ec o
is e ical and he eloci y indica es ha he cu e is going o
u n up and each he uns able egime.
Fo he equa ion
z ðα; Þ¼uðα; Þ¼ðu1ðα; Þ;u2ðα; ÞÞ;
ha is,
a: ∂αz1ðαÞ>0i α≠0;b:∂αz1ð0Þ¼0;
c:∂αz2ð0Þ>0;d:∂αu1ð0Þ<0;
o analy ic unc ions z1ðαÞand z2ðαÞsuch ha zðαÞsa is ies
he a c-cho d condi ion. He e we conside he pe iodic case
(being analogous o an open cu e anishing a in ini y).
We assume ha zðαÞis a smoo h odd cu e sa is ying he
p ope ies a,b, and c. Di e en ia ing he exp ession 10 o
he ho izon al componen o he eloci y, a α¼0, i yields
ð∂αu1Þð0Þ¼Zπ
−π
½cosðz1ðβÞÞð∂αz1ðβÞÞ2
þsinðz1ðβÞÞ∂2
αz1ðβÞ∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ
−Zπ
−π
sinðz1ðβÞÞ∂αz1ðβÞ½sinðz1ðβÞÞ∂αz1ðβÞ
−sinhðz2ðβÞÞð∂αz2ð0Þ−∂
αz2ðβÞÞ∕½ðcoshðz2ðβÞÞ
−cosðz1ðβÞÞÞ2dβ:
In eg a ion by pa s p o ides
Zπ
−π
½sinðz1ðβÞÞ∂2
αz1ðβÞ∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ
¼−Zπ
−π
cosðz1ðβÞÞ½ð∂αz1ðβÞÞ2∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ
þZπ
−π
sinðz1ðβÞÞ∂αz1ðβÞ½sinðz1ðβÞÞ∂αz1ðβÞ
þsinhðz2ðβÞÞ∂αz2ðβÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2dβ:
The e o e, i is easy o ob ain ha
ð∂αu1Þð0Þ¼∂αz2ð0ÞZπ
−π
½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ
−cosðz1ðβÞÞÞ2∂αz1ðβÞdβ
¼2∂αz2ð0ÞZπ
0
½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ
−cosðz1ðβÞÞÞ2∂αz1ðβÞdβ
[11]
Exp ession 11 allows us o de e mine he sign o ð∂αu1Þð0Þ.
One could ake
z1ðβÞ¼−sinðβÞþβ
4756 ∣www.pnas.o g/cgi/doi/10.1073/pnas.1101518108 Cas o e al.
and cons uc he unc ion z2ðβÞin he ollowing way: Le β1,
β2,β3, and β4be eal inc easing numbe s less han π. We pick
z2ðβÞ≤0 o β2<β<π,z2ðβÞ<c<0 o β2<β<β4, and
z
2ðβÞa smoo h unc ion wi h he ollowing p ope ies
a: z
2ðβÞis odd;b:ð∂βz
2Þð0Þ>0;
c: z
2ðβÞ>0i β∈ð0;β1Þ;d:z

2ðβÞ<0i β∈ðβ1;β2:
Also, z
2ðβÞis 2π-pe iodic. Fo z2ðβÞ¼bz
2ðβÞ,0≤β≤β2, and
b>0, he eloci y sa is ies
ð∂αu1Þð0Þ<2ð∂αz2Þð0Þ
×Zβ1
0
½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ
−cosðz1ðβÞÞÞ2∂αz1ðβÞdβ
þZπ
β3
½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ
−cosðz1ðβÞÞÞ2∂αz1ðβÞdβ
¼2ð∂αz2Þð0ÞZβ1
0
½sinðz1ðβÞÞ sinhðbz
2ðβÞÞ∕½ðcoshðbz
2ðβÞÞ
−cosðz1ðβÞÞÞ2∂αz1ðβÞdβþA;
whe e A<0. The cons an bla ge enough yields ð∂αu1Þ
ð0Þ<0.
Rec i ying he cu e on he in e al ½β2;β3, i is easy o ob ain
a smoo h cu e. Finally, con ol ing wi h he hea ke nel he
e ical componen , he cu e zðαÞis app oxima ed by an
analy ic one.
4. Fou h, wi h he ini ial da a ound in 3 and no assump ion
on he R-T condi ion, we use a modi ica ion o Cauchy–
Kowalewski heo ems (24, 25) o show ha he e exis s an ana-
ly ic solu ion o he Muska p oblem in some in e al ½−T;T
o a small enough T>0. He e we a e o ced o change sub-
s an ially he me hod in e . 26 because, in his case, he cu e
canno be pa ame e ized as a g aph, so we ha e o deal wi h
he a c-cho d condi ion. Then, wi h X g >0, a scale o Banach
spaces gi en by eal unc ions ha can be ex ended analy ically
on he complex s ip S ¼ αþiζ∈C:jζj< gwi h no m
‖ ‖ ¼∑
Zj ðαi Þj2dαþZj∂4
α ðαi Þj2dα;
and z0ðαÞa cu e sa is ying he a c-cho d condi ion and
z0ðαÞ∈X 0 o some 0>0, we p o e he exis ence o a ime
T>0and 0< < 0so ha he e is a unique solu ion o he
Muska p oblem in Cð½−T;T;X Þ. This esul allows us o
ind solu ions ha do no sa is y he R-T bu sh ink he s ip
o analy ici y. We ex end Eq. 10 as ollows:
z ðαþiζ; Þ¼Gðzðαþiζ; ÞÞ;
wi h
GðzÞðα; Þ¼ðρ2−ρ1Þ
4π
×ZT
sinðz1ðα; Þ−z1ðα−β; ÞÞð∂αzðα; Þ−∂
αzðα−β; ÞÞ
coshðz2ðα; Þ−z2ðα−β; ÞÞ −cosðz1ðα; Þ−z1ðα−β; ÞÞ dβ:
Fo 0< 0< and he open se Oin S gi en by
O¼ z∈X :‖z‖ <R; ‖FðzÞ‖L∞ðS Þ<R2g;[12]
he unc ion G o G:O→X 0is a con inuous mapping and
he e is a cons an CR(depending on Ronly) such ha
‖GðzÞ‖ ≤CR
− 0‖z‖ ;[13]
‖Gðz2Þ−Gðz1Þ‖ 0≤CR
− 0‖z2−z1‖ ;[14]
and
sup
αþiζ∈S ;β∈T
jGðzÞðαþiζÞ−GðzÞðαþiζ−βÞj ≤CRjβj;[15]
o z;zj∈O. Fo ini ial da a z0∈X 0sa is ying a c-cho d, we can
ind a 0< 0
0< 0and a cons an R0such ha ‖z0‖ 0
0<R0and
½coshðz0
2ðαþiζÞ−z0
2ðαþiζ−βÞÞ
−cosðz0
1ðαþiζÞ−z0
1ðαþiζ−βÞÞ∕ð‖β‖2Þ>1
R2
0
;[16]
o αþiζ∈S 0
0. We ake 0< < 0
0and R0<R o de ine he
open se Oas in [12]. The e o e we can use he classical me hod
o successi e app oxima ions:
znþ1ð Þ¼z0þZ
0
GðznðsÞÞds;
o G:O→X 0and 0< 0< . We assume by induc ion ha
‖zk‖ ð Þ<R; and ‖FðzkÞ‖L∞ðS Þð Þ<R
o k≤nand 0< <Twi h T¼minðTA;TCK Þand TCK he
ime ob aining in he p oo s in e s 24 and 25. We ge
‖znþ1‖ ð Þ<R ha ollows using [13 and 14]. The ime TAis
o yield ‖Fðznþ1Þ‖L∞ðS Þð Þ<R. Then, using he induc ion
hypo hesisand[15],wecancon ol hequan i y aking0<TA<
ðR−2
0−R−2ÞðC2
Rþ2R0CRÞ−1.
5. Fi h, all he esul s abo e allow us o p o e ha he e is a
nonemp y se o ini ial da a in H4sa is ying he a c-cho d
and R-Tcondi ions, such ha he solu ion o he Muska p o-
blem eaches he uns able egime: The R-T becomes s ic ly
nega i e on a nonemp y in e al. We pick ini ial da a as in
3. We apply he local-exis ence esul in 4 o ge an analy ic
solu ion zðα; Þon ½−T;T. Then we conside a ime 0<δ<T
and a cu e ωε
δðα; Þ, sol ing he Muska p oblem wi h ini ial
da um zðα;−δÞþηϵ
δðαÞ. The unc ion ηϵ
δhas a small H4
no m, i.e.,
‖ωε
δð·;−δÞ−zð·;−δÞ‖H4¼‖ηϵ
δ‖H4≤ε:
The ime δis small enough so ha ωε
δðα;−δÞsa is ies R-T:
ðρ2−ρ1Þ∂αðωε
δÞ1ðα;−δÞ>0. Then we apply he local-exis ence
esul in 1 ha ωε
δðα; Þbecomes analy ic o some ime −δ< .
Wi h 2, we assu e he exis ence and analy ici y o he solu ion
e en i ∂αðωε
δÞ1ðα; Þ≤0 o some ime . Then, we show ha
bo h solu ions a e close in he H4 opology as ime e ol es.
We can apply o ωε
δ he local-exis ence esul in 4 i i is needed.
Then, wi h δand εsmall enough, we ind he desi ed esul .
3. Tu ning Wa e Wa es
In his sec ion, we p o e o he wa e wa e p oblem (ρ1¼0and
[1–3and 5]) ha wi h ini ial da a gi en by a g aph ðα; 0ðαÞÞ,
he in e ace eaches a egime in ini e ime whe e i only
can be pa ame e ized as zðα; Þ¼ðz1ðα; Þ;z2ðα; ÞÞ; o α∈R, wi h
∂αz1ðα; Þ<0 o α∈I, a nonemp y in e al. The e o e he e
exis s a ime whe e he solu ion o he ee bounda y p oblem
epa ame e ized by ðα; ðα; ÞÞ sa is ies ‖ α‖L∞ð Þ¼∞.
Theo em 3.1. The e exis s a nonemp y open se o ini ial da a
ðα; 0ðαÞÞ, wi h 0∈H5, such ha in ini e ime  he solu ion o
Cas o e al. PNAS ∣Ma ch 22, 2011 ∣ ol. 108 ∣no. 12 ∣4757
MATHEMATICS
he wa e wa es p oblem (ρ1¼0and [1–3and 5]) gi en by
ðα; ðα; ÞÞ sa is ies ‖ α‖L∞ð Þ¼∞. The solu ion can be con inued
o >
as zðα; Þwi h ∂αz1ðα; Þ<0 o α∈I, a nonemp y in e al.
In o de o p o e his heo em, we conside a cu e zðαÞ∈H5
wi h he same p ope ies as in poin 3 o he p e ious sec ion.
Then, we pick zðα; Þ¼zðαÞand ωðα; Þ¼−∂αz
2ðαÞas a da um
o he ini ial alue p oblem. I is easy o ind he same p ope ies
o he eloci y, because he angen ial di ec ion does no a ec
he e olu ion. Picking he app op ia e cðα; Þand applying he
local-exis ence esul in e . 18 (no e ha in his case i is no
necessa y analy ici y, jus H5 egula i y), he e exis s a solu ion
o he wa e wa es p oblem wi h zðα; Þ∈Cð½ −δ; þδ;H5Þ,
ωðα; Þ∈Cð½ −δ; þδ;H4Þ, and δ>0small enough. Then,
he ini ial da um ðz0ðαÞ;ω0ðαÞÞ¼ðα; 0ðαÞ;ω0ðαÞÞ is gi en by
ðzðα; −δÞ;ωðα; −δÞÞ.
4. Muska B eakdown
In his sec ion, we show ha he e exis s a smoo h ini ial da a in
he s able egime o he Muska p oblem such ha he solu ion
u ns o he uns able egime and la e i b eaks down. The ou line
o he p oo is o cons uc a cu e in he uns able egime which is
analy ic excep in a single poin . We show ha , as we e ol e back-
wa d in ime, he cu e becomes analy ic and is as close as we
desi ed (in he Hk opology wi h kla ge enough) o he cu e
om pa 3 o Sec ion 2.
He e we will wo k in he pe iodic se ing and will conside he
equa ion
∂ zðζ; Þ¼Zw∈Γþð Þ
sinðz1ðζ; Þ−z1ðw; ÞÞ
coshðz2ðζ; Þ−z2ðw; ÞÞ −cosðz1ðζ; Þ−z1ðw; ÞÞ
×ð∂ζzðζ; Þ−∂
ζzðw; ÞÞdw; [17]
whe e ζ∈Ωð Þ,
Ωð Þ¼ ζ∈C∕2kπ:jℑζj<hðℜz; Þg;
hðx; Þis a posi i e pe iodic unc ion wi h pe iod 2πand smoo h
o ixed ime , and
Γð Þ¼ ζ∈C∕2kπ:ζ¼xþihðx; Þg:
This equa ion is equi alen o [1,2, and 4] o holomo phic
unc ions.
In o de o p o e he esul , we will need he ollowing heo-
em:
Theo em 4.1. Le hðx; Þbe a posi i e, smoo h, and pe iodic unc ion
wi h pe iod 2π o ixed ime ∈½ 0−δ; 0. Le zðx; 0Þbe a cu e
sa is ying he ollowing p ope ies:
•z1ðx; 0Þ−xand z2ðx; 0Þa e pe iodic wi h pe iod 2π;
•zðζ; 0Þis eal o ζ eal;
•zðζ; 0Þis analy ic in ζ∈Ωð 0Þ;
•zðζ; Þ∈HkðΓð 0ÞÞ wi h ka la ge enough in ege .
•Complex a c-cho d condi ion:
jcoshðz2ðζ; 0Þ−z2ðw; 0ÞÞ −cosðz1ðζ; 0Þ−z1ðw; 0ÞÞj
≥½jjℜðζ−wÞjj þ jℑðζ−wÞj2;
o ζ,w∈Ωð 0Þ, whe e ‖x‖¼dis anceðx;2kπÞ:
•Gene alized Rayleigh–Taylo condi ion: RTðζ; 0Þ>0, whe e
RTðζ; Þ¼ℜ−2π∂ζz1ðζ; Þ
ð∂ζz1ðζ; ÞÞ2þð∂ζz2ðζ; ÞÞ2ð1þi∂xhðℜζ; ÞÞ−1
þℑPV ZwΓþð Þ
½sinðz1ðζ; Þ
−z1ðw; ÞÞ∕½coshðz2ðζ; Þ−z2ðw; ÞÞ
−cosðz1ðζ; Þ−z1ðw; ÞÞdw þi∂ hðζ; Þ
×ð1þi∂xhðℜζ; ÞÞ−1:
Then, o small enough δ, he e exis s a solu ion o Eq. 17 in he
ime in e al ∈½ 0−δ; 0, sa is ying
•z1ðx; Þ−xand z2ðx; Þa e pe iodic wi h pe iod 2π;
•zðζ; Þis eal o ζ eal;
•zðζ; Þis analy ic in ζ∈Ωð 0Þ;
•zðζ; Þ∈HkðΓð ÞÞ wi h ka la ge enough in ege .
Now, le zðx; Þbe he solu ion o he Muska p oblem wi h
zðx;0Þ¼z0ðxÞ, whe e z0ðxÞis he pa icula ini ial da a om pa 3
o he Sec ion 2. We shall de ine his solu ion as he unpe u bed
solu ion. Le us deno e he Rayleigh–Taylo unc ion
σ0
1ðx; Þ≡−2π∂xz1ðx; Þ
ð∂xz1ðx; ÞÞ2þð∂xz2ðx; ÞÞ2:
No ice he minus sign in he igh -hand side o he p e ious
exp ession. One can check he ollowing p ope ies o his
Rayleigh–Taylo unc ion:
1. σ0
1ð·; Þis analy ic on xþiy:x∈T;jyj≤cbgwi h jσ0
1ðxþiy; Þj
≤C, o all xþiy as abo e and o all ≤½0;τ;
2. σ0
1ð0;0Þis eal o x∈T, ∈½0;τ;
3. σ0
1has a p io i bounded Ck0no m as a unc ion o
ðx; Þ∈T×½0;τ(k0la ge enough);
4. σ0
1ð0;0Þ¼0;
5. ∂xσ0
1ð0;0Þ¼0;
6. ∂2
xσ0
1ð0;0Þ¼−c2<0;
7. ∂ σ0
1ð0;0Þ¼c1>0.
In his se ing, we de ine he ollowing weigh unc ions
hðx; Þ¼A−1ðτ2− 2ÞþðA−1−ðτ− ÞÞ sin2x
2 o ∈½τ2;τ:
[18]
ℏðx; Þ¼1
4ðA−1τ2þA−1sinx
2ÞþA−2τ þA sinx
2
∈½0;τ2;[19]
wi h x∈T. Fi s we choose he pa ame e s Ala ge enough and
hen τsmall enough, hen one can show ha
σ0
1ðx; Þþ∂ hðx; Þ−A1
2hðx; Þ≥cτ2 o x∈T; ∈½τ2;τ[20]
and
σ0
1ðx; Þþ∂ ℏðx; Þ−A1
2ℏðx; Þ≥1
2A−2τ o x∈T; ∈½0;τ2:[21]
The inequali ies 20 and 21 a e one o he main ing edien s o he
p oo o he ollowing esul s.
Theo em 4.2. Le zðx; Þbe a solu ion o he Muska equa ion in he
in e al ∈½0;τ. Le hðx; Þand ℏðx; Þas in he exp essions 18 and
19, and ka la ge enough in ege . Assume ha zðx; Þsa is ies
4758 ∣www.pnas.o g/cgi/doi/10.1073/pnas.1101518108 Cas o e al.

•z1ðx; Þ−xand z2ðx; Þa e pe iodic wi h pe iod 2π;
•zðζ; Þis eal o ζ eal;
•zðζ; Þis analy ic in ζ∈Ωð Þ;
•zðζ; Þ∈HkðΓð ÞÞ wi h ka la ge enough in ege .
•Complex a c-cho d condi ion:
jcoshðz2ðζ; Þ−z2ðw; ÞÞ −cosðz1ðζ; Þ−z1ðw; ÞÞj
≥½‖ℜðζ−wÞ‖þjℑðζ−wÞj2;
o ζ,w∈Ωð Þ.
He e, in he de ini ion o Ωð Þand Γð Þ, we use hðx; Þi ∈½τ2;τ
and ℏðx; Þi ∈½0;τ2. Then
1
2
d
d Zw∈Γþð Þ
j∂k
ζzðζ; Þ−∂
k
ζzðζ; Þj2dℜζ≥−CðAÞλ2;
i ∈½τ2;τ
Zw∈Γþð Þ
j∂k
ζzðζ; Þ−∂
k
ζzðζ; Þj2dℜζ≤λ2
and λ≤τ50.
In addi ion,
1
2
d
d Zw∈Γþð Þ
j∂k
ζzðζ; Þ−∂
k
ζzðζ; Þj2dℜζ≥−CðAÞτ−1λ2;
i ∈½0;τ2
Zw∈Γþð Þ
j∂k
ζzðζ; Þ−∂
k
ζzðζ; Þj2dℜζ≤λ2
and λ≤τ50.
This heo em implies ha o all γ>0 he e is ε>0such ha
Zw∈Γþð Þ
j∂k
ζzðζ; Þ−∂
k
ζzðζ; Þj2dℜζ≤γ
o ∈½0;τi
Zw∈Γþð Þ
j∂k
ζzðζ;τÞ−∂
k
ζzðζ;τÞj2dℜζ≤ε
and zðx; Þsa is ies he equi emen s o he heo em.
Lemma 4.3. Le zðx; Þbe a solu ion o he Muska p oblem sa is ying
he equi emen s o Theo em 4.2 and close enough o he unpe -
u bed solu ion in ∈½0;τ. Le hðx; Þand ℏðx; Þbe as in [18]
and [19] wi h a sui able choice o Aand τ. Then zðx; Þsa is ies
he gene alized Rayleigh–Taylo condi ion in ∈½0;τ. In pa icula ,
he unpe u bed solu ion sa is ies he gene alized Rayleigh–Taylo
condi ion in ∈½0;τ
Theo ems 4.1 and 4.2 and Lemma 4.3 allow us o achie e he
desi ed esul . Indeed we can choose a cu e zðx;τÞsuch ha
Zζ∈Γ
j∂k
ζzðζ;τÞ−∂
k
ζzðζ;τÞj2dℜζ≤ε;
wi h 0<ε<ε0(ε0small enough), sa is ying he gene alized
Rayleigh–Taylo condi ion by Lemma 4.3 and sa is ying he es
o he hypo hesis o Theo em 4.1. Because hð0;τÞ¼0,zðx; Þis
allowed o be nonanaly ic a x¼0[maybe zðx;τÞ∈HkðTÞbu
zðx;τÞ∉Hkþ1ðTÞ]. By Theo em 4.1, he e is a solu ion zðx; Þ, ana-
ly ic in Ωð Þ, o some in e al ∈½τ−δ;τwi h small enough δ
and o all ε. By Theo em 4.2, we can choose εsmall enough
in such a way ha , by Lemma 4.3, zðx;τ−δÞsa is ies he gene al-
ized Rayleigh–Taylo condi ion. Then we can go u he he ime
τ−δ. I e a ing his a gumen , we ind we can ex end zðx; Þ o be a
solu ion o he Muska p oblem, analy ic in Ωð Þ o all ∈½0;τ
and as close as we wan o he unpe u bed solu ion.
ACKNOWLEDGEMENTS A.C., D.C., and F.G. we e pa ially suppo ed by G an
MTM2008-03754 o he Minis e io de Ciencia e Inno ación (MCINN) (Spain)
and G an S G-203138CDSIF o he Eu opean Resea ch Council. C.F. was pa -
ially suppo ed by Na ional Science Founda ion (NSF) G an DMS-0901040
and O ice o Na al Resea ch G an ONR00014-08-1-0678. F.G. was pa ially
suppo ed by NSF G an DMS-0901810. M.L.-F. was pa ially suppo ed by
G an s MTM2008-03541 and MTM2010-19510 o he MCINN (Spain).
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MATHEMATICS